# Long-term Causal Inference Under Persistent Confounding via Data Combination

arXiv: https://arxiv.org/abs/2202.07234 · HTML: https://mlsanity.com/paper/2202.07234/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 95 references that are in the map.
- First posted on arXiv 15 Feb 2022.

## Builds on (most cited first, 30 of 40)

- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Deep Generalized Method of Moments for Instrumental Variable Analysis](https://mlsanity.com/paper/1905.12495/) (2019)
- [Minimax Estimation of Conditional Moment Models](https://mlsanity.com/paper/2006.07201/) (2020)
- [Off-Policy Evaluation in Partially Observable Environments](https://mlsanity.com/paper/1909.03739/) (2019)
- [Removing Hidden Confounding by Experimental Grounding](https://mlsanity.com/paper/1810.11646/) (2018)
- [Semiparametric proximal causal inference](https://mlsanity.com/paper/2011.08411/) (2020)
- [Proximal Causal Learning with Kernels: Two-Stage Estimation and Moment Restriction](https://mlsanity.com/paper/2105.04544/) (2021)
- [Causal inference methods for combining randomized trials and observational studies: a review](https://mlsanity.com/paper/2011.08047/) (2020)
- [Causal Inference Under Unmeasured Confounding With Negative Controls: A Minimax Learning Approach](https://mlsanity.com/paper/2103.14029/) (2021)
- [Proximal Reinforcement Learning: Efficient Off-Policy Evaluation in Partially Observed Markov Decision Processes](https://mlsanity.com/paper/2110.15332/) (2021)
- [Deep Proxy Causal Learning and its Application to Confounded Bandit Policy Evaluation](https://mlsanity.com/paper/2106.03907/) (2021)
- [On the role of surrogates in the efficient estimation of treatment effects with limited outcome data](https://mlsanity.com/paper/2003.12408/) (2020)
- [A confounding bridge approach for double negative control inference on causal effects](https://mlsanity.com/paper/1808.04945/) (2018)
- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [Combining multiple observational data sources to estimate causal effects](https://mlsanity.com/paper/1801.00802/) (2018)
- [Combining Observational and Experimental Datasets Using Shrinkage Estimators](https://mlsanity.com/paper/2002.06708/) (2020)
- [Using Experiments to Correct for Selection in Observational Studies](https://mlsanity.com/paper/2006.09676/) (2020)
- [Kernel Methods for Unobserved Confounding: Negative Controls, Proxies, and Instruments](https://mlsanity.com/paper/2012.10315/) (2020)
- [Targeting for long-term outcomes](https://mlsanity.com/paper/2010.15835/) (2020)
- [The Variational Method of Moments](https://mlsanity.com/paper/2012.09422/) (2020)
- [Adaptive Combination of Randomized and Observational Data](https://mlsanity.com/paper/2111.15012/) (2021)
- [Proximal Learning for Individualized Treatment Regimes Under Unmeasured Confounding](https://mlsanity.com/paper/2105.01187/) (2021)
- [Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference](https://mlsanity.com/paper/2104.02929/) (2021)
- [Elastic Integrative Analysis of Randomized Trial and Real-World Data for Treatment Heterogeneity Estimation](https://mlsanity.com/paper/2005.10579/) (2020)
- [Proximal Causal Inference for Complex Longitudinal Studies](https://mlsanity.com/paper/2109.07030/) (2021)
- [Proximal mediation analysis](https://mlsanity.com/paper/2109.11904/) (2021)
- [Propensity Score Methods for Merging Observational and Experimental Datasets](https://mlsanity.com/paper/1804.07863/) (2018)
- [Data fusion methods for the heterogeneity of treatment effect and confounding function](https://mlsanity.com/paper/2007.12922/) (2020)

## Built on by (most cited first, 27 of 27)

- [Combining Observational and Randomized Data for Estimating Heterogeneous Treatment Effects](https://mlsanity.com/paper/2202.12891/) (2022)
- [Detecting hidden confounding in observational data using multiple environments](https://mlsanity.com/paper/2205.13935/) (2022)
- [Long-term Causal Effects Estimation via Latent Surrogates Representation Learning](https://mlsanity.com/paper/2208.04589/) (2022)
- [Estimating long-term causal effects from short-term experiments and long-term observational data with unobserved confounding](https://mlsanity.com/paper/2302.10625/) (2023)
- [Meta-Learners for Partially-Identified Treatment Effects Across Multiple Environments](https://mlsanity.com/paper/2406.02464/) (2024)
- [Policy Learning for Balancing Short-Term and Long-Term Rewards](https://mlsanity.com/paper/2405.03329/) (2024)
- [Constructing Confidence Intervals for Average Treatment Effects from Multiple Datasets](https://mlsanity.com/paper/2412.11511/) (2024)
- [Non-parametric identifiability and sensitivity analysis of synthetic control models](https://mlsanity.com/paper/2301.07656/) (2023)
- [The Odyssey of Commonsense Causality: From Foundational Benchmarks to Cutting-Edge Reasoning](https://mlsanity.com/paper/2406.19307/) (2024)
- [Spillover Detection for Donor Selection in Synthetic Control Models](https://mlsanity.com/paper/2406.11399/) (2024)
- [Estimating Long-term Heterogeneous Dose-response Curve: Generalization Bound Leveraging Optimal Transport Weights](https://mlsanity.com/paper/2406.19195/) (2024)
- [Long-Term Individual Causal Effect Estimation via Identifiable Latent Representation Learning](https://mlsanity.com/paper/2505.05192/) (2025)
- [Estimating Heterogeneous Treatment Effects by Combining Weak Instruments and Observational Data](https://mlsanity.com/paper/2406.06452/) (2024)
- [On the Role of Surrogates in Conformal Inference of Individual Causal Effects](https://mlsanity.com/paper/2412.12365/) (2024)
- [Nonparametric Heterogeneous Long-term Causal Effect Estimation via Data Combination](https://mlsanity.com/paper/2502.18960/) (2025)
- [Long-term Causal Inference via Modeling Sequential Latent Confounding](https://mlsanity.com/paper/2502.18994/) (2025)
- [Identification and Estimation for Nonignorable Missing Data: A Data Fusion Approach](https://mlsanity.com/paper/2311.09015/) (2023)
- [A General Framework for Off-Policy Learning with Partially-Observed Reward](https://mlsanity.com/paper/2506.14439/) (2025)
- [Pareto-Optimal Estimation and Policy Learning on Short-term and Long-term Treatment Effects](https://mlsanity.com/paper/2403.02624/) (2024)
- [Online Data Collection for Efficient Semiparametric Inference](https://mlsanity.com/paper/2411.03195/) (2024)
- [Optimal Policy Adaptation under Covariate Shift](https://mlsanity.com/paper/2501.08067/) (2025)
- [Identification and Estimation of Long-Term Treatment Effects with Monotone Missing](https://mlsanity.com/paper/2504.19527/) (2025)
- [Transferring Causal Effects using Proxies](https://mlsanity.com/paper/2510.25924/) (2025)
- [Integrating RCTs, RWD, AI/ML and Statistics: Next-Generation Evidence Synthesis](https://mlsanity.com/paper/2511.19735/) (2025)
- [Testing Effect Homogeneity and Confounding in High-Dimensional Experimental and Observational Studies](https://mlsanity.com/paper/2602.19703/) (2026)
- [Orthogonal Learner for Estimating Heterogeneous Long-Term Treatment Effects](https://mlsanity.com/paper/2604.00915/) (2026)
- [Representation Learning for Sample-Efficient CATE Estimation by Leveraging Multiple Outcomes](https://mlsanity.com/paper/2609.06294/) (2026)

## Abstract (arXiv)

We study the identification and estimation of long-term treatment effects when both experimental and observational data are available. Since the long-term outcome is observed only after a long delay, it is not measured in the experimental data, but only recorded in the observational data. However, both types of data include observations of some short-term outcomes. In this paper, we uniquely tackle the challenge of persistent unmeasured confounders, i.e., some unmeasured confounders that can simultaneously affect the treatment, short-term outcomes and the long-term outcome, noting that they invalidate identification strategies in previous literature. To address this challenge, we exploit the sequential structure of multiple short-term outcomes, and develop three novel identification strategies for the average long-term treatment effect. We further propose three corresponding estimators and prove their asymptotic consistency and asymptotic normality. We finally apply our methods to estimate the effect of a job training program on long-term employment using semi-synthetic data. We numerically show that our proposals outperform existing methods that fail to handle persistent confounders.